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Optimal Integration of Battery Systems in Grid-Connected Networks for Reducing Energy Losses and CO_{2} Emissions

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## Abstract

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_{2}emissions, for which it formulates a mathematical model that considers the constraints associated with the operation of GCNs in a distributed generation environment that includes BS and variable power generation related to photovoltaic (PV) distributed generation (DG) and demand. As solution strategies, three different master–slave methodologies are employed that are based on sequential programming methods, with the aim to avoid the implementation of commercial software. In the master stage, to solve the problem regarding the location and the type of batteries to be used, parallel-discrete versions of the Montecarlo method (PMC), a genetic algorithm (PDGA), and the search crow algorithm (PDSCA) are employed. In the slave stage, the particle swarm optimization algortihm (PSO) is employed to solve the problem pertaining to the operation of the batteries, using a matrix hourly power flow to assess the impact of each possible solution proposed by the master–slave methodologies on the objective functions and constraints. As a test scenario, a GCN based on the 33-bus test systems is used, which considers the generation, power demand, and CO

_{2}emissions behavior of the city of Medellín (Colombia). Each algorithm is executed 1000 times, with the aim to evaluate the effectiveness of each solution in terms of its quality, standard deviation, and processing times. The simulation results obtained in this work demostrate that PMC/PSO is the master–slave methodology with the best performance in terms of solution quality, repeatability, and processing time.

## 1. Introduction

#### 1.1. General Context

_{2}emissions and the energy losses associated with energy transport, directly affecting the quality of electrical services and the life conditions of GCN users [1]. With the aim to mitigate this issue, in the last years, grid operators and researchers have been given the task of integrating environmentally friendly distributed energy resources, which involves the smart integration and operation of renewable energy resources, battery systems, capacitor banks, and static compensators, among others [2,3]. BS have been the most widely installed and studied distributed energy resources in recent years, as they allow managing the energy of the grid, mitigating the variability of renewable energy resources, and improving the economic, technical, and environmental conditions of the network (i.e., the reduction of energy purchasing costs, energy losses, and CO

_{2}emissions, as well as voltage profile improvements, among others) [4]. In turn, GCNs are the most widely studied electrical grids [5], as they are the most developed networks around the world and the ones with the most technical problems (i.e., energy losses, voltage profile limit violations, and line overloadability, among others) [6]. Furthermore, since they are located in cities and towns, fossil fuel-based generation contributes to CO

_{2}emissions in their corresponding regions, affecting the health of their inhabitants. Based on the above, the authors of this work focus on the problem regarding the selection, location, and operation of BS in GCNs, with the aim to reduce energy losses and CO

_{2}emissions.

#### 1.2. State of the Art

_{2}emissions and the improvement of the technical aspects of the grid as objective functions. In this work, the authors compare the best and average solutions obtained by means of the proposed methodology against those of other works reported in the literature. However, they use a mono-nodal grid as a test system and do not analyze the reported processing times and standard deviation values. In addition, this work does not analyze the effectiveness of the studied solution methods regarding processing times and repeatability. The authors of [17] also locate and size BS in a mono-nodal grid. In this work, they describe the mathematical formulation for the integration of BS and other energy resources that compose the electrical grid, (i.e., distributed generators and loads, among others). This study considers economical and technical indicators as objective functions, comparing the results obtained with those of other works reported in the literature. The main problem with these works is associated with the fact that multi-nodal GCNs are more widely used in real life, which implies several constraints related to voltage profiles and line currents in their mathematical formulations, thus increasing the complexity of the problem. However, the analyzed works offer important information on the variable power generation and demand of electrical grids, as well as data related to the implementation of smart optimization methods in the problem regarding the optimal operation and location of BS in GCNs.

_{2}emissions and energy power losses by using a non-linear mathematical formulation, which was solved with the GAMS software. Nevertheless, the results obtained were not compared with those of other works reported in the literature. By using specialized software, the authors of [27] obtained the optimal location and operation scheme of BS in an electrical grid, solving a second-order cone programming model with the MATLAB CVX tool. This work used different test systems, but comparison methods were not considered. In [28], a mixed-integer linear programming model was proposed for representing the problem pertaining to the location and operation of BS in electrical networks, in order to reduce investments and CO

_{2}emissions from fossil fuel-based power generation. This work considered different kinds of batteries and load curves, with the aim to analyze the effect of different devices on the operation of an electrical system. Note that the works reported in the literature for reducing CO

_{2}emissions with the operation of BS use specialized software for solving the proposed mathematical models. This is explained by the fact that the mathematical models used are still being built and validated. Therefore, there is a need to propose mathematical models that guarantee the correct operation of the grid when this environmental indicator is used, as well as solutions based on sequential programming methods that avoid the use of specialized software.

_{2}emissions. Furthermore, these new methodologies must guarantee the best results in terms of technical and environmental indices, with the aim to obtain resilient strategies that consider the needs of the GCN and the community while avoiding the implementation of specialized software, which increases the costs and complexity of the solution methodologies [29]. Furthermore, these methodologies must be compared against other approaches, with the aim to identify the solution methodology with the best performance in terms of solution quality, repeatability, and processing times.

#### 1.3. Scope and Main Contributions

_{2}emissions, the authors of this paper propose a complete mathematical formulation of the problem regarding the selection, location, and operation of BS in GCN for reducing energy losses and CO

_{2}emissions. All this is conducted while including all constraints related to the electrical network (global power balance and line current and voltage profile limits), conventional and distributed generators (power limits), and BS (discharging and charging power limits and state of charge limits). Furthermore, with the aim to use the high-performance methodologies reported in the literature for solving electrical problems similar to that studied herein, three discrete versions of some optimization methods were used in the master stage. The first of these is the parallel Montecarlo algorithm (PMC) [30], which employs a random search process to find the solution with the best performance while taking advantage of parallel processing, using all workers in the computer to reduce processing times. Moreover, following the suggestions made in the literature, this paper generated two parallel-discrete versions of two continuous optimization methods: the genetic (PDGA) and crow search (PDCSA) algorithms. For the slave stage, the particle optimization algorithm proposed in [31] was adapted. This algorithm was developed for operating batteries in direct current (DC) grids, with no application or validation in alternating current (AC) networks. By combining the three optimization methods proposed in master stage and PSO, it was possible to obtain three new master–slave methodologies for solving the problem under study, namely, PMC/PSO, PDGA/PSO, and PDCSA/PSO (hereinafter called PMC, PDGA, and PDCSA for the sake of simplicity). In addition, with the aim to evaluate the objective function and constraints related to each solution offered by these strategies, aiming for the shortest processing times and the best convergence, this study used the hourly power flow matrix based on successive approximations, as proposed in [23], which allows considering variations in distributed generation and power demand. As a test scenario, an adapted version of the 33-bus test system was used, which represents the technical and environmental conditions of the city of Medellín (Colombia), while considering the operation of three photovoltaic (PV) generators with maximum power point tracking, which is the traditional way to operate renewable generation technology in this country. Finally, to evaluate the performance of the proposed solution methodologies, 1000 executions of each one were carried out, with the aim to evaluate the minimum and average solution, as well as the standard deviation and average processing times. This analysis allowed selecting the optimization methodology with the best performance for solving the problem regarding the optimal integration and operation of BS in GCNs.

#### 1.3.1. Academic Contributions

- A mathematical model for the optimal integration of BS in GCN whose objective function is the reduction of energy losses and CO
_{2}emissions, observing all of the constraints that represent the operation of a GCN in an environment of variable distributed generation and power demand. - A discrete codification for the problem regarding the location and selection of BS.
- A continuous codification for the problem regarding the operation of the batteries located in the GCN.
- Three new master–slave strategies (PMC, PDGA, and PDCSA) for solving the problem regarding the optimal integration of BS in GCNs.
- The identification of PMC as the master–slave strategy with the best performance in terms of solution quality and its repeatability and processing times for solving the problem under study. This optimization methodology could be used in future works for the sake of comparison, with the aim to obtain methodologies with a better performance.

#### 1.3.2. Industrial Applications

- A mathematical formulation that allows the grid operators to quantify energy losses and CO
_{2}emissions before and after considering the integration of BS in GCNs. - An effective and fast optimization method based on sequential programming, which allows determining the location and operation scheme of multiple batteries within the grid, with the purpose of reducing the energy losses and CO
_{2}emissions while observing all operating constraints.

#### 1.4. Paper Organization

## 2. Mathematical Formulation

_{2}emissions in GCNs. Furthermore, this section describes all of the constraints related to the technical limitations of the devices that make up the grid, as well as the operation limits associated with voltage profiles and line currents.

## 3. Proposed Solution Methodologies

#### 3.1. Master–Slave Methodology and Codifications Used

_{2}emissions, as well as the constraints that make up the problem, it is necessary to determine the power flow for the different periods, with the aim to analyze the effect of the power generated and demanded by the loads, PV generators, and batteries installed in the GCN. After evaluating each period of time and obtaining the values associated with the objective function and constraints, this information is summarized with the aim to evaluate the effect of the batteries on the grid during an average day of operation. A power flow evaluation within a multi-hour scenario is known as an hourly power flow. In this work, a matrix hourly power flow based on successive approximations (MHPF) was selected, given the excellent results reported in [23]. Algorithm 1 describes this method:

Algorithm 1: Algorithm proposed for the matrix hourly power flow based on successive approximations |

#### 3.2. Master Stage

#### 3.2.1. Parallel Montecarlo Method (PMC)

Algorithm 2: Algorithm proposed for the PMC |

_{2}emissions), satisfying the set of constraints that compose the problem under study. Therefore, each individual of the population must be processed by the slave stage, which implies long processing times. This is addressed by using parallel processing, with the aim to evaluate multiple individuals at the same time. After evaluating the objective function values of the population, the best solution is included in the elite list, a process that is repeated iteration to iteration until the maximum number of iterations is reached. When this occurs, the best solution of the elite list is selected as the solution to the problem. This solution contains the location, selection, and operation of the batteries for a day which yields the lowest objective function value.

#### 3.2.2. Parallel-Discrete Genetic Algorithm (PDGA)

Algorithm 3: Iterative process of the PDGA |

#### 3.2.3. Parallel-Discrete Crow Search Algorithm (PDCSA)

Algorithm 4: Iterative process of the PDCSA |

#### 3.3. Slave Stage

Algorithm 5: Iterative process proposed for the PSO used in the slave stage |

## 4. Test Scenarios and Considerations

_{2}emissions from conventional generators (electrical grid) of the city of Medellín (Colombia), as well as PV-DGs and three different kinds of lithium-ion batteries (types A, B, and C), with different power capacities and charge and discharge times [27]. Lithium-ion batteries are a type of rechargeable battery which uses the reversible reduction of lithium ions to store energy. They are highly used in the literature because they have a higher energy density, a higher efficiency, and a longer useful life. Traditional lead acid batteries allow 1500 life-cycles, while lithium battery technology offers a duration of up to 2500 [40].

_{2}emissions associated with the generators located in the grid, this work considered 0.1644 kg of CO

_{2}per kWh as the emissions factor for the conventional generators, as well as a value of 0 kg of CO

_{2}per kWh for the PV-DGs, as this kind of generator does not emit greenhouse gases or release carbon-based pollutants when producing energy [43]. The authors of this paper acknowledge the environmental impact of constructing PV modules, just as well as the fact that this technology does not affect environmental conditions when used for generating energy.

## 5. Simulation Results

_{2}emissions. All simulations were carried out in the Matlab 2023 software, on a Dell Workstation with an Intel(R) Xeon(R) E5-1660 v3 3.0 GHz processor, 16 GB DDR4 RAM, and a 480 GB 2.5″ solid state hard drive, with 8 workers running on Windows 11 Pro. All simulations were executed 1000 times in order to evaluate performance in terms of the average solution and processing times, as well as regarding the standard deviation.

_{2}emissions were analyzed without considering the BS installed in the grid. Thus, the base scenario involved variable power demand and the PV distributed generators operating in maximum power point tracking (MPPT) mode (Figure 5). This scenario obtained values of 2484.5746 kWh for energy losses and 9887.4082 kg of CO

_{2}(9.88 Ton) for the CO

_{2}emissions.

_{2}. The average reduction in this environmental index (after 100 executions) was 20.2162 kg of CO

_{2}. With respect to the base case, these values correspond to reductions of 0.2294% and 0.2044%, respectively. As in the case of the $Eloss$, considering a year of operation, the optimization methods would achieve a total reduction of 7.37 Ton of CO

_{2}on average, thus demonstrating the environmental importance and effectiveness of the integration, selection, and smart operation of BS in GCNs.

_{2}emissions is illustrated in Figure 8b. In this case, the BS were located at buses 25, 30, and 10 (all of them type A). The batteries follow the same dynamics: they start at 50% SOC, discharging all batteries until hour 9. They start the charging process from this hour until hour 16, and they discharge until hour 24, achieving the final SOC (50%). The batteries satisfy the state-of-charge limits at all times.

## 6. Conclusions and Future Work

_{2}emissions. As solution methods, three different master–slave methodologies were proposed. In the master stage, the PMC, PDGA, and PDCSA were employed for selecting and locating three different BS types in a GCN. Furthermore, the slave stage used PSO for the operation of the batteries, as well as a matrix hourly power flow to calculate the objective functions and evaluate the technical and operating constraints involved in the mathematical formulation. Finally, with the aim to identify the solution methodology with the best performance, each algorithm was executed 1000 times, analyzing the best and average solutions, the standard deviation, and the processing times. The 33-bus test system was used for validation, which was adapted to represent the power demand and PV power generation of the city of Medellín (Colombia). This city constitutes an excellent test scenario, given its high energy losses and CO

_{2}emissions levels, as well as its excellent conditions for PV generation (this kind of renewable energy is widely used in the city). In this paper, the PV-DGs were considered to operate in maximum power point tracking mode, with the aim to make the best out of this resource.

_{2}emissions for an average operation day, respectively. These reductions are equivalent to 42.89 MWh and 7.37 Ton of CO

_{2}in a year of operation. These values are significant for the operation of grid-connected electrical distribution systems, as they imply commercial and environmental benefits. In addition, the proposed solution methodologies reported a low standard deviation, with average values of 0.3220% and 0.01124% for energy losses and CO

_{2}emissions, respectively. Moreover, in a problem as demanding as the integration of BS in GCN, the implementation of a matrix hourly power flow based on successive approximations allowed reducing the processing times by about 68%, with values of 6792.92 and 7440.969347 s regarding energy losses and CO

_{2}emissions. With this information, it can be concluded that these strategies allow solving the problem regarding the selection, location, and operation of multiple BS in a GCN in about 2 h, which allows electrical operators to evaluate multiple generation and demand scenarios, as well as different electrical systems, in reduced times, which is very important for public bidding processes.

_{2}emissions.

_{2}emissions. Finally, the mathematical formulation could include economical indicators with regard to the cost of the BS, by using multi-objective functions that consider the improvement of technical, economical, and environmental indicators.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BS | Battery systems. |

GCN | Grid-connected network. |

PV | Photovoltaic. |

DG | Distributed generator. |

PMC | Parallel-discrete version of the Montecarlo method. |

PDGA | Parallel-discrete version of the genetic algorithm. |

PDSCA | Parallel-discrete version of the search crow algorihm. |

PSO | Particle swarm optimization algorithm. |

CO_{2} | Carbon dioxide. |

GAMS | General Algebraic Modeling System. |

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**Figure 3.**Codification used to find the operation scheme of the batteries selected and located by the master stage.

**Figure 6.**Reductions obtained by the proposed master–slave methodologies regarding the base case: (

**a**) minimum and average reductions of the objective function and (

**b**) standard deviation (percentage) and processing times.

**Figure 7.**Percent reductions obtained by the PMC with regard to the other comparison methods: (

**a**) in the minimum and average objective function values, (

**b**) in the standard deviation and processing times.

**Figure 9.**Values obtained by the PMC regarding energy loss reductions: (

**a**) line current (A), (

**b**) bus voltage (p.u.).

**Figure 10.**Values obtained by the PMC regarding CO

_{2}emissions reductions: (

**a**) line current (A), (

**b**) bus voltage (p.u.).

Line l | Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}\phantom{\rule{0.222222em}{0ex}}\left(\mathsf{\Omega}\right)$ | ${\mathit{X}}_{\mathbf{ij}}\phantom{\rule{0.222222em}{0ex}}\left(\mathsf{\Omega}\right)$ | ${\mathit{P}}_{\mathit{j}}\phantom{\rule{0.222222em}{0ex}}$(kW) | ${\mathit{Q}}_{\mathit{j}}\phantom{\rule{0.222222em}{0ex}}$(kVAr) | ${\mathit{I}}_{\mathbf{ij}}^{max}\phantom{\rule{0.222222em}{0ex}}$(A) |
---|---|---|---|---|---|---|---|

1 | 1 | 2 | 0.0922 | 0.0477 | 100 | 60 | 385 |

2 | 2 | 3 | 0.4930 | 0.2511 | 90 | 40 | 355 |

3 | 3 | 4 | 0.3660 | 0.1864 | 120 | 80 | 240 |

4 | 4 | 5 | 0.3811 | 0.1941 | 60 | 30 | 240 |

5 | 5 | 6 | 0.8190 | 0.7070 | 60 | 20 | 240 |

6 | 6 | 7 | 0.1872 | 0.6188 | 200 | 100 | 110 |

7 | 7 | 8 | 1.7114 | 1.2351 | 200 | 100 | 85 |

8 | 8 | 9 | 1.0300 | 0.7400 | 60 | 20 | 70 |

9 | 9 | 10 | 1.0400 | 0.7400 | 60 | 20 | 70 |

10 | 10 | 11 | 0.1966 | 0.0650 | 45 | 30 | 55 |

11 | 11 | 12 | 0.3744 | 0.1238 | 60 | 35 | 55 |

12 | 12 | 13 | 1.4680 | 1.1550 | 60 | 35 | 55 |

13 | 13 | 14 | 0.5416 | 0.7129 | 120 | 80 | 40 |

14 | 14 | 15 | 0.5910 | 0.5260 | 60 | 10 | 25 |

15 | 15 | 16 | 0.7463 | 0.5450 | 60 | 20 | 20 |

16 | 16 | 17 | 1.2890 | 1.7210 | 60 | 20 | 20 |

17 | 17 | 18 | 0.7320 | 0.5740 | 90 | 40 | 20 |

18 | 2 | 19 | 0.1640 | 0.1565 | 90 | 40 | 40 |

19 | 19 | 20 | 1.5042 | 1.3554 | 90 | 40 | 25 |

20 | 20 | 21 | 0.4095 | 0.4784 | 90 | 40 | 20 |

21 | 21 | 22 | 0.7089 | 0.9373 | 90 | 40 | 20 |

22 | 3 | 23 | 0.4512 | 0.3083 | 90 | 50 | 85 |

23 | 23 | 24 | 0.8980 | 0.7091 | 420 | 200 | 85 |

24 | 24 | 25 | 0.8960 | 0.7011 | 420 | 200 | 40 |

25 | 6 | 26 | 0.2030 | 0.1034 | 60 | 25 | 125 |

26 | 26 | 27 | 0.2842 | 0.1447 | 60 | 25 | 110 |

27 | 27 | 28 | 1.0590 | 0.9337 | 60 | 20 | 110 |

28 | 28 | 29 | 0.8042 | 0.7006 | 120 | 70 | 110 |

29 | 29 | 30 | 0.5075 | 0.2585 | 200 | 600 | 95 |

30 | 30 | 31 | 0.9744 | 0.9630 | 150 | 70 | 55 |

31 | 31 | 32 | 0.3105 | 0.3619 | 210 | 100 | 30 |

32 | 32 | 33 | 0.3410 | 0.5302 | 60 | 40 | 20 |

Type | Capacity (kWh) | Charge Time (h) | Discharge Time (h) |
---|---|---|---|

A | 1000 | 4 | 4 |

B | 1500 | 4 | 4 |

C | 2000 | 5 | 5 |

Minimum Solution | Average Solution | |||
---|---|---|---|---|

Method | Eloss (kWh) | Emissions (Ton CO_{2}) | Eloss (kWh) | Emissions (Ton CO_{2}) |

PMC | 2350.8270 | 9864.5471 | 2358.9454 | 9866.72854 |

PDGA | 2336.0684 | 9862.8580 | 2347.9000 | 9865.1420 |

PDCA | 2354.5459 | 9864.7264 | 2367.0639 | 9867.1920 |

Standard deviation (%) | Processing time (s) | |||

Method | Eloss | Emissions | Eloss | Emissions |

PMC | 0.2391 | 0.0112 | 75.0580 | 75.4806 |

PDGA | 0.3829 | 0.0147 | 7477.4304 | 7338.0373 |

PDCA | 0.4592 | 0.0143 | 6792.9205 | 7440.9693 |

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## Share and Cite

**MDPI and ACS Style**

Grisales-Noreña, L.F.; Montoya, O.D.; Perea-Moreno, A.-J.
Optimal Integration of Battery Systems in Grid-Connected Networks for Reducing Energy Losses and CO_{2} Emissions. *Mathematics* **2023**, *11*, 1604.
https://doi.org/10.3390/math11071604

**AMA Style**

Grisales-Noreña LF, Montoya OD, Perea-Moreno A-J.
Optimal Integration of Battery Systems in Grid-Connected Networks for Reducing Energy Losses and CO_{2} Emissions. *Mathematics*. 2023; 11(7):1604.
https://doi.org/10.3390/math11071604

**Chicago/Turabian Style**

Grisales-Noreña, Luis Fernando, Oscar Danilo Montoya, and Alberto-Jesus Perea-Moreno.
2023. "Optimal Integration of Battery Systems in Grid-Connected Networks for Reducing Energy Losses and CO_{2} Emissions" *Mathematics* 11, no. 7: 1604.
https://doi.org/10.3390/math11071604